Simplex preserving transformation in $\Bbb R^2$

Question

I am looking for a transformation that preserves the vertices of a triangle located at the $\Bbb R^2$ simplex and moves a point at $(x_1,y_1)$ to a point inside the triangle $(u_1,v_1)$.

I am hoping for an equation that builds a $2 \times 2$ matrix with a known mapping for a single point, let's say source $(0.1,0.1)$ and a destination $(0.1,0.2)$.

Does anybody know how to construct such a transformation?

Answer 1

It is not possible with a matrix. A matrix in $R^2$ will always represent a rotation/translation/scale/reflection transformation, at most. You would need a conformal transformation, which is more general but which conserves angles.